We propose a novel island nucleation and growth model explicitly including transient (ballistic) mobility of the monomers deposited at rate F, assumed to be in a hot precursor state before thermalizing. In limiting regimes, corresponding to fast (diffusive) and slow (ballistic) thermalization, the island density N obeys scaling N∝F(α). In between is found a rich, complex behavior, with various distinctive scaling regimes, characterized by effective exponents α(eff) and activation energies that we compute exactly. Application to N(F,T) of recent organic-molecule deposition experiments yields an excellent fit.